Identifying Conic Sections and their parts (focus, directrix) - Hyperbola - Parabola - Ellipse - General form and eliminating the xy term by rotation

Math Help > Geometry > Circles, Conic Sections > Conic Sections

A conic section is the intersection of a plane with a cone. can be defined: as the locus of all points P, such that the ratio of the distance from P to a fixed point (the focus), to the distance from P to a fixed line (the directrix), is constant. The value of this constant is known as the eccentricity, e. The eccentricity can also be interpreted as the fraction of the distance from the center to a vertex along the semimajor axis at which the focus lies.

General form of a conic section: ax² + 2hxy + by² + 2gx + 2fy + c = 0

The general form of the equation of a conic section is ax² + 2hxy + by² + 2gx + 2fy + c = 0

If h²<ab, the conic is an ellipse; the plane intersects just one side of the cone at a greater angle than the cone's central angle.

If h²>ab, the conic is a hyperbola; the plane intersects both sides of the cone at a smaller angle than the cone's central angle.

If h²=ab, the conic is a parabola; the plane intersects just one side of the cone at exactly the same angle as the cone's central angle.

Eliminating the xy term

In order to eliminate the xy term from the equation

ax² + 2hxy + by² + 2gx + 2fy + c = 0,

the axes should be rotated counter-clockwise through an angle equal to

½ arctan( 2h/(a-b) )

Hyperbola (e>1)

Hyperbola parameters

2a is the length of the transverse axis, which connects the vertices;
2b is the length of the conjugate axis, which is perpendicular to the transverse axis;
(h,k) is the "center" of the hyperbola, where the axes intersect.
The eccentricity, e, of the hyperbola is defined using b=a sqrt(e²-1), where e>1.
Solving this equation for e,
ae = sqrt(a² + b²)
e = sqrt(1 + b²/a²)

A hyperbola that opens left and right is (x-h)²/a² - (y-k)²/b² = 1

The vertices are at (h-a,k) and (h+a,k)

The foci are at (h-ae,k) and (h+ae,k),
which can also be expressed
(h-sqrt(a²+b²),k) and
(h+sqrt(a²+b²),k)

A hyperbola that opens up and down is (y-k)²/b² - (x-h)²/a² = 1

The vertices are at (h,k-a) and (h,k+a)

The foci are at (h,k-ae) and (h,k+ae),
which can also be expressed
(h,k-sqrt(a²+b²)) and
(h,k+sqrt(a²+b²))

A "rectangular hyperbola" is xy=k

Parabola (e=1)

Parabola parameters

The eccentricity, e, of a parabola is defined as 1.

A parabola with a vertical axis of symmetry is
y = ax² + bx + c,
where a is not equal to zero.

The axis of symmetry of the parabola is x = -b/(2a). The vertex is

(x, ax² + bx + c) =
(-b/(2a), a(-b/(2a))²+b(-b/(2a))+c) =
(-b/(2a), b²/(4a)-(b²/(2a))+c) =
(-b/(2a), -(b²/(4a))+c) =
(-b/(2a), -(b²-4ac)/(4a))

If the equation of the parabola is rewritten (y-k) = a(x-h)² then the vertex is (h,k), the focus is (h,k+1/(4a)) and the directrix is y=k-1/(4a).

The latus rectum is the chord of the parabola parallel to the directrix that passes through its focus,
y=k+1/(4a).
Since a parabola is the set of points equidistant from the directrix and focus, then the point of intersection of the latus rectum and the parabola has distance 1/(2a) from the directrix and 1/(2a) from the focus. Thus the length of the latus rectum is 2/(2a)=1/a.

A parabola with its axis parallel to the x axis is
x = ay² + by + c,
where a is not equal to zero.

The axis of symmetry of the parabola is y = -b/(2a). The vertex is

(ay² + by + c, y) =
( a(-b/(2a))²+b(-b/(2a))+c, -b/(2a)) =
( b²/(4a)-(b²/(2a))+c, -b/(2a)) =
( -(b²/(4a))+c, -b/(2a)) =
( -(b²-4ac)/(4a), -b/(2a))

If the equation of the parabola is rewritten (x-h) = a(y-k)² then the vertex is (h,k), the focus is (h+1/(4a),k) and the directrix is x=h-1/(4a).

The latus rectum is the chord of the parabola parallel to the directrix that passes through its focus, x=h+1/(4a).

The length of the latus rectum is 1/a.

Ellipse (0≤e<1)

Ellipse parameters

2a is the length of the major axis, which is the longer axis;
2b is the length of the minor axis;
(h,k) is the "center" of the ellipse, where the axes intersect.
The eccentricity, e, of the ellipse is defined using b=a sqrt(1-e²), where 0≤e<1.
Solving this equation for e,
ae = sqrt(a² - b²)
e = sqrt(1 - b²/a²)

An ellipse with a horizontal major axis is
(x-h)²/a² + (y-k)²/b² = 1,
where a>b

The foci are at (h-ae,k) and (h+ae,k),
which can also be expressed
(h-sqrt(a² - b²),k) and
(h+sqrt(a² - b²),k)

An ellipse with a vertical major axis is
( x-h)²/b² + ( y-k)²/a² = 1,
where a>b

The foci are at (h,k-ae) and (h,k+ae),
which can also be expressed
(h,k-sqrt(a² - b²)) and
(h,k+sqrt(a² - b²))

Internet references

Mathwords: foci of an ellipse, focus of a parabola, foci of a hyperbola

Planet Math: conic section, and the Dandelin sphere

Wikipedia: Rotation of Axes

Mudd Math Fun Facts: eccentricity of conic sections

Mathworld: eccentricity -- can be interpreted as the fraction of the distance along the semimajor axis at which the focus lies.

Related pages in this website

Rotation of axes
Cyclic quadrilaterals -- bunch of theorems related to inscribing polygons in conic sections

The webmaster and author of this Math Help site is Graeme McRae.